I think I get what you're asking. If it's a party where the adults are expected to stay, how do you know if the host is planning on just one adult per child or both, and if only one, which one is the host asking for?
I'm honestly not sure how to determine how many adults are expected from the household... If it's at someone's home, which is spacious, it might be that both adults are fine, whereas if you have a space-limited venue just one adult might be more convenient. Probably when I RSVP'd I would be careful to say exactly who all was coming ("Me, Bob, and Billy," instead of just "us") so the host could prepare.
However, I kind of feel like it doesn't matter much which adult goes, if you suspect only one is expected, or only one is available. At least, I don't like the idea of the host saying, "Bob, you and Billy are invited to my child's birthday party. However, your wife Betty can't come. So if you're out of town, just forget the whole thing." You know? That seems pretty rude to me (barring something extreme of course). So I would say that even though Sharon addressed the invitation specifically to DH, if he's not available you're free to attend with your child instead.
Now if DH and your child were out of town, it might be weird to attend the party on your own, or it might not, depending on your relationship with the host and the birthday child. In this particular case it sounds like it might be weird, but I could see an aunt attending her nephew's birthday party, even if her own kids were away at camp or something.